On adiabatic and non-adiabatic improvements of the mean-field approximation for fermion- or (pseudo-)spin-boson interaction

Starting at the tight-binding electron-phonon Hamiltonian for a linear diatomic chain (half-filled Peierls chain with extrinsic gap), being a special realization of the fermion- or (pseudo)spin-boson interaction, we examine some improvements of the usual MFA: an improved phonon trial function with (anti-)symmetrized coherent states; a generalized Hartree-Fock decoupling with anomalous averages after transforming the electron-phonon interaction to an effective electron-electron one; a pure bosonic treatment using the Holstein-Primakoff transformation. Systematic improvements at the ground state are obtained. But for finite temperatures the formalism soon becomes quite elaborate. Nevertheless, these different approaches provide valuable insights into various aspects of this interaction problem. Furthermore, the unifying features of various particle-particle interactions are emphasized and the connection between solid-state (physics) and (molecular) quantum chemistry is mentioned. (author)